$Φ$-Entropy Inequality and Invariant Probability Measure for SDEs with Jump

Abstract: By using the $\Phi$-entropy inequality derived in \cite{Wu, Ch} for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump L\'evy processes. The semigroup $\Phi$-entropy inequality for SDEs driven by Poisson point processes as well as a sharp result on the existence of invariant probability measures are also presented.